I have hunted crows since before I was a teenager, and have been calling them about that long.
I like a good hand call. I usually get settled in, wait a few minutes so any crows in the area forget about me, then give a simple 'caw caw caw' greeting call... "Where's the party, boys?"
If I can get one to answer me, I just talk back to him. I'll get a little aggressive by drawing out the call... 'ca-a-a-a-aw, 'ca-a-a-a-aw' and putting some feeling in it. That usually ruffles the feathers of one of the regulars and he'll come to chase the intruder off. I'll shut up between calls, too, and be very still.
Lots of time the first crow will make a silent pass over your site to check things out. If it spots you the jig is up as it'll leave while calling the alarm call. When I'm rifle hunting I'm real still, and listening hard for the wingbeats of a silently approaching 'point man'. If I can spot it before it spots me, I have a good chance of taking it after the crow lands and before it spots me. With a rifle, you'll usually only get one shot, but sometimes you can get a couple depending on the firearm (quieter guns fool more crows).
When I'm shotgun hunting, I've had the most success by getting in thick low brush in open woods near a stream or creek, repeating the same basic strategy but moving on to a crow fight (calling very aggressively with lots of crow cussing). Many's the time I've gone through more than a box of shells, shooting wildly in self-defense as I get crows mobbing me, zipping through the trees within a few yards to see what crow is getting the stuffing beat out of him. It only helps if you knock a couple down without killing them immediately and their fluttering or calling just excites the rest of the flock beyond all reason. Now that kind of hunting is the stuff that keeps you coming back for more.